Distributions pp 153-176 | Cite as

Fractional Integration and Differentiation

  • J. J. Duistermaat
  • J. A. C. Kolk
Part of the Cornerstones book series (COR)


In this chapter we deal with “complex powers of the operator \(\frac{d}{{dx}},\)” a concept already found in the posthumous works of Riemann and elaborated by Marcel Riesz in the 1930s and 1940s. The relevant article [18] is lengthy, but with the help of a little distribution theory and complex analysis, all results can readily be proved.We will also deal with Riesz’s treatment of the wave operator \(\Box = \partial _t^2 -\triangle_{x}\) in arbitrary dimension; thus we will obtain, among other things, a fundamental solution of \(\Box.\)


Cauchy Problem Fundamental Solution Analytic Continuation Radon Measure Fractional Integration 
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Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Mathematical InstituteUtrecht UniversityUtrechtThe Netherlands

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